Análisis de Estabilidad de Pequeña Señal Para SVC y HVDC

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'rk actions on Power Systems, vol 6 NO 3, August 1991 1147

S L IA L L S I G N A L S T A B I L I T Y P RO GR Ai zl A N A LY S I SOF SVC A N D HL DC I N A C P OW E R S Y S T E M S

M G LaubyY. WOnQ P . undurS. Ar abi G. J . Roqer sSeni or Member, I EEE

Ontar i o Hydro700 Uni ver si t y Ave

Tor ont o, Ont ari oM5G x6

Canada

Abst r act - HVdc t r ansm ssi on l i nks and SVCs havecontr ol l abl e charact er i s t i cs w th potent i al f oraff ecti ng system stabi l i ty. To study these ef f ect sand to desi gn thei r cont r ol l ers f or i mprovi ng syst emstabi l i ty, t here i s a need for thei r r epresentati on i nsmal l si gnal st abi l i t y programs as wel l as t i mesi mul at i on progr ams. I n t hi s paper, we addr ess t hef ormul at i on of dc l i nk and SVC model s and t hei rcontr ol l ers f o r smal l s i gnal s tabi l i ty.

Sever al exampl es are studi ed t o show t he capabi l i t yand appl i cati on of t he smal l si gnal dc l i nk and SVCmodel s. The smal l si gnal r esul t s are ver i f i ed by t i medomai n si mul at i on r esul t s of t he same study cases.

Key Wor ds:

HVdc Tr ansm ssi on, St at i c VAr Compensat or, Smal lSi gnal St abi l i t y, Ei genval ues, Modal Anal ysi s.

I NTRODUCTI ON

Hi gh vol t age di r ect cur r ent l i nks are bei ng usedextensi vel y i n i nter connect ed power syst emsworl dw de. They ar e used bot h t o t r ansf er power,economcal l y, over l ong di st ances, and t o t r ansf erpower bet ween t wo sys t ems whi ch do not r un i nsynchr oni sm I n Nort h Ameri ca t he t otal capaci t y OfHVdc l i nks i n 1987 was over 14, 000 MW [ l ] w t hi ncreases pl anned.

The real power f l ow t hr ough HVdc l i nks, and ther eact i ve power absor bed by t hem may be c ont r ol l ed byvar yi ng the f i r i ng angl es of t hei r ac/ dc convert ers.Besi des pr ovi di ng a means of power t r ansm ssi on, HVdcl i nks can al so be used f or cont r ol l i ng t he dampi ng oft he osci l l ati ons between synchr onous generators,i nherent i n power syst ems.

Because of t he f ast cont r ol l abi l i t y and hi gh powercapaci t y of HVdc l i nks, i ncorr ect l y desi gned contr ol smay degrade t he per f ormance of t he system I t i si mpor t ant to be abl e to appl y syst emati c desi gnt echni ques t o HVdc cont rol s. I t i s al so i mport ant t obe abl e t o use suf f i ci ent l y detai l ed power systemmodel s, so t hat bot h l ocal and gl obal ef f ect s of t hedc l i nk contr ol s can be studi ed.

Model s of HVdc l i nks have been avai l abl e, f or use i nt i me domai n si mul at i on programs, f or some t i me. Themost ext ensi ve t r eat ment of dc model l i ng i n t hi s t ypeof si mul ati on i s t hat devel oped under EPRI Proj ectRP1964-4 [2]. The use of t r ansi ent st abi l i t y pr ogr ams

91 WM 217- 0 PWRSby the I EEE Power Syst emEngi neeri ng Comm t t ee of

t he I EEE Power Engi neeri ng Soci ety f or present ati onat t he I EEE/ PES 1991 W nter Meet i ng, New Yor k,New Yor k, Februar y 3 - 7 1991. Manuscri pt subm t t edAugust 30 1990; made avai l abl e f or pri nt i ngJ anuar y 22 1991.

A paper r ecommended and appr oved

Seni or Member, I EEEEPRI

3412 Hi l l vi ewAve.Pal o Al t o, CA 94303

U. S. A.

Fel l ow I EEE

f or t he devel opment of HVdc l i nk contr ol s has l ed to at r i al - and- err or procedure usi ng r epeat ed si mul ati onst o eval uat e the ef f ect of desi gn changes.

Stat i c VAr Compensator s ( SVCs) ar e of t en associ atedw t h HVdc l i nks suppl yi ng power i nt o rel ati vel y weaksystems. They al so have pot ent i al t o provi de f ast VArsuppor t i n other ci r cumst ances. Li ke HVdc l i nks, t heymay be used to i mprove dampi ng i n syst emosci l l ati ons.

The st udy of power system osci l l ati ons and t he desi gnof cont r ol s to ai d i n t he dampi ng of t heseosci l l ati ons i s best done usi ng modal anal ysi st echni ques on a l i near i zed model of t he power syst emI t i s i mpor t ant t o be abl e t o model al l devi ces whi chhave a si gni f i cant i mpact on t he stabi l i t y of t heoveral l system Thi s paper descr i bes t he f ormul ati onof HVdc l i nk and SVC model s f or use i n power syst emsmal l si gnal st abi l i t y studi es. Such model s areavai l abl e i n t he Smal l Si gnal St abi l i t y Program ( SSSP)package devel oped by Ontar i o Hydro under EPRIRP2447-1 [3]. Each model i s i n t he f orm of al i neari zed st at e space system w t h i t s out put t hechange i n cur r ent i nj ect i on i nt o the tr ansm ssi onnet work. A si mpl e syst em model havi ng both an HVdcl i nk and an SVC i s used to i l l ustr at e the capabi l i t yand appl i cat i on of t he model s. Fr equencyr esponse capabi l i t y of t he package, as r el ated tot hese model s, i s al so demonst r ated. The smal l s i gnalstabi l i t y anal ysi s resul ts are compared w t h t hei rt i me si mul at i on counter par t s. Modes obtai ned byspect r al anal ysi s of t he ti me si mul ati on r esul t s arever y cl ose to t hose obt ai ned by smal l si gnal st abi l i t yanal ysi s.

HI GH VOLTAGE DI RECT CURRENT TRANSM SSI ON

Most HVdc l i nks t hat are i n servi ce can be model l ed as

t wo t erm nal s havi ng one cont r ol l ed r ect i f i er and onecont r ol l ed i nvert er as l i nks to t he ac system So

f ar, each dc l i nk has been desi gned on an i ndi vi dualbasi s and, al t hough t he basi c ac/dc conver t ers arest andar d, t he method of t hei r cont r ol and the nat ureof the cont r ol var i abl es used vari es consi derabl y f r omi nstal l at i on to i nstal l ati on. hTearl y every uni t i nservi ce has cont rol f aci l i ti es, addi ti onal to thebasi c dc curr ent and vol t age cont r ol l ers, desi gned t oensur e t hat t he i nt erconnected syst em i s st abl ef ol l ow ng bot h smal l and l arge di st ur bances. ' Thef or mul at i on of an adequate model f or smal l si gnalst abi l i t y must t ake i nto account the need f orf l exi bi l i t y i n the contr ol s as wel l as ther epr esent ati on of t he basi c acti on of t he l i nk.

I n many t r ansi ent stabi l i t y pr ogr ams, dc l i nks arer epr esented by a f unct i onal model i n whi ch t hedynam cs of t he dc l i ne and the conver t er f i r i ng angl econt r ol s are consi der ed t o be i nstantaneous. Such

model s are general l y t oo rest ri cti ve f or use i n smal lsi gnal st abi l i t y anal ysi s and the model pr esent ed her ei ncl udes the dynamcs of t he l i ne and the conver t ercontr ol l ers. Nevert hel ess, i t i s possi bl e to use thedynam c model f or a f unct i onal r epr esentat i on bymaki ng assumpti ons regardi ng t he cont r ol l er and t hel i ne dynam cs t hat are descr i bed l at er. The gener al

0885-8950/91/0700-1147$01.00 1991



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f orm of any devi ce model i n SSSP [ 3 ] i sr

d x l d t = A x + B S A v T + B ,AV,

A i T = C x - Y, Av T - Y, AV,

wher e,

x i s a vector of state vari abl es,

A i s the st at e mat r i x,

Bs i s a mat ri x descri bi ng t he i nt er acti onbetween t he ac t erm nal vol t age changes AV

B i s a matr i x descri bi ng t he i nt eracti onbetween t he changes i n r emote bus vol t ages AVrand the st ates,

Ai T i s the vector of ac curr ent i nj ect i onchanges at t he ac term nal s,

C i s t he mat ri x descri bi ng t he i nf l uence of t hest ates on t he curr ent i nj ecti ons,

Ys i s t he eff ect i ve sel f admt t ance of t hedevi ce, and

Yr descr i bes t he i nf l uence of t he changes i nr emot e bus vol t ages on the curr ent i nj ecti ons.

Each model i s f or med usi ng t he component connect i onmet hod [ 4 ] whi ch al l ows great f l exi bi l i ty i n modelstr ucture.

The HVdc model uses t he normal cont i nuousapproxi mati on to t he dc conver t er [ 5 ] l i neari zedabout an i ni t i al condi ti on determ ned f rom ani ni t i at i ng l oad f l ow The dc l i ne i s model l ed as a

T ci r cui t and i ncl udes t he r esi st ance and i nduct anceof smoothi ng i nductor s. The l i ne char gi ng may bei gnored i f r equi r ed, but t he i nduct i ve component oft he l i ne must be r epr esent ed. The ful l l i ne modelr equi r es t hr ee states, t he cur r ent s at t he t woconver t ers and the capaci t or vol t age.

The l i ne model negl ecti ng char gi ng r equi r es onl y asi ngl e state, t he dc l i ne curr ent. Two separat econt rol l ers ar e provi ded, one for t he rect i f i er andone for t he i nvert er. The out put s of t he cont rol l er s

are t he respect i ve convert er f i r i ng angl es.The HVdc model of SSSP w t h user- def i ned cont rol s i sshown i n Fi gur e 1. The pr ogr am has al so a st andar dmodel opt i on f or use i n i ni t i al pl anni ng st udi es orwhen data or more speci f i c cont r ol s are not avai l abl e.

DC Net work and Conver t er Model g

The st at e space equat i ons of t he dc net work may best ated i n mat r i x f ormas:

dx Idt = A,$&+ B d c ~E O B,A a (3)

wher e,

Xdc i s a vect or of dc network st ate var i abl es,

Adc i s t he dc network s t at e matr i x, and

Bdc and Ba ar e matr i ces descr i bi ng t hei nter act i on bet ween t he commutat i on vol t agechanges AEo and conver t er f i r i ng angl e changesAa r especti vel y and t he rate of change of xdc*

The convert er s ar e model l ed usi ng t hei r wel l - knownst eady- st ate al gebr ai c equati ons [ 5 ] , whi ch may best at ed i n matr i x f or mas:

E

INPUTS

= ?Em m

MODUATDNcotrm s

.._.._......... .....

WEFITERCoNlRoLS

FIGU RE1 USER DEFINEDHVDC UNK MODEL

A i, = AKx&+YYdcOL\EOAK, A a

wher e,

( 4 )

AK and AKa are matr i ces descr i bi ng t hei nf l uence of t he dc net work st ates and conver t erf i r i ng angl e changes r especti vel y on the conver t erac cur r ent changes Ai ac, and,

Ydco i s t he ef f ecti ve commutat i ng bus adm t t ancemat r i x.

I n order t o be abl e to model conver t er t r ansf ormersw th ter t i ary w ndi ngs, the ac l ow tensi on ( LT)conver t er buses are used as t he i nj ecti on t erm nal s,w t h the fol l ow ng rel ati onshi p t o the commutati ng busvol t age changes:

A E AVa: +Z, A i,

where, 2 i s t he commutat i on i mpedance mat r i x.

Cont r ol Model s

Two t ypes of cont r ol model have been provi ded. One i scompl et el y def i ned by hard code and i s i ntended f oruse i n i ni t i al pl anni ng studi es bef ore det ai l edi nf ormati on i s avai l abl e on the pl anned i nstal l at i on.The other i s user def i ned and i s i nt ended f or det ai l edmodel l i ng of dc l i nks w t h known cont rol l er st ructure.

The i ni t i al pl anni ng model i s based on t ypi cal( equi di st ant fi ri ng angl e) convert er cont rol s w t hcurr ent cont rol at t he recti f i er, and dc vol t agecont rol at t he i nvert er. Modul ati on of the recti f i erand i nver t er cont r ol s usi ng t hei r r especti ve ac busf requency i s al so provi ded. I t s formul at i on i ssi m l ar t o t hat of t he user def i ned cont r ol model t hati s di scussed bel ow

gser - Def i ned Cont r ol Model

I n addi t i on to t he dc l i ne and conver t er model s, basi cl i near and non- l i near bui l di ng bl ocks are avai l abl ef or model const ructi on. The l i near bl ocks provi de al lt ypes of t ransf er f uncti on t hat are usual l y used i nany cont rol l er . The non- l i near bl ocks provi de f or t henon- l i near f uncti ons t hat occur i n dc cont rol l er s.

T



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such as di vi si on (eg, power di vi ded by vol t age t ocal cul ate current) , mul t i pl i cat i on (eg, vol t age ti mesmargi n angl e t o cal cul ate margi n area), and cosi nei nver se ( f or i t s l i neari zati on ef f ect on the overal lcl osed l oop system.

The mai n i nter f ace t o the ac system i s t hr ough theconver t er ac bus vol t age magni t ude and angl e, curr entmagni t ude, and real and r eact i ve power. Si m l arquant i t i es associ ated w t h t wo r emote buses areavai l abl e as cont r ol i nput s. Convert er dc quant i t i essuch as cur r ent , vol t age, and i nvert er exti nct i onangl e are al so avai l abl e f or use as f eedback si gnal s

i n di f f erent cont rol l ers. Ac bus f r equency can beobt ai ned by usi ng a washout ( part i al di f f erent i ati on)bl ock w t h bus angl e i nput . The t i me const ant of t hewashout t hen r epr esent s t he f r equency t r ansducer t i meconst ant . No pr ogr amm ng i s r equi r ed t o i mpl ement auser def i ned model and i t s const r ucti on i sstr ai ght f orward, once a bl ock di agram of thecont r ol l er has been establ i shed.

The f ormul ati on of t he cont r ol s i s as f ol l ows:

d x c / d t = A c x c t B c A V b ( 6 )

(7)

A U b = A K c ~ y b + B K c ~ u , 8)

A a = C K C a y b (9)

A U = CGx,+BG,Ai,+BG,av,+BG,av,t BG,Aa ( 1 0 )

wher e,xc i s a vect or of contr ol l ers st at e vari abl es,

AC i s t he cont rol l ers state matr i x,

BC i s a matr i x descri bi ng the i nt eracti onbet ween the contr ol bl ock i nput changes Aub,and the cont r ol l ers st at es,

Cc and Dc are matr i ces descr i bi ng t hei nf l uence of t he cont rol l ers s t ates and cont rolbl ock i nput changes respecti vel y on the cont r olbl ock out put changes Ayb,

AKc and BKc ar e matr i ces descr i bi ng t hei nf l uence of t he contr ol bl ock out put changes andcont r ol l ers i nput changes Auo r espect i vel y ont he contr ol bl ock i nput changes,

CKc i s a matr i x descr i bi ng t he i nf l uence of t hecont r ol bl ock output changes on t he conver t erf i r i ng angl e changes,

CG i s a matr i x descr i bi ng t he i nf l uence of t he dcnet work st ates on the cont r ol l ers i nput changes,

BG1 and BG are mat r i ces descr i bi ng t hei nf l uence of t he convert er ac curr ent changes andconver t er f i r i ng angl e changes r especti vel y on t hecont r ol l ers i nput changes, and

BG2 and BG3 are mat r i ces descr i bi ng t hei nfl uence of t he conver t er ac LT bus vol t agechanges AV and r emot e bus vol t age changesAvr r espect i vel y on t he cont rol l er s i nputchanges.

Fi nal l y, t he general f orm of t he model i s obt ai ned byel i mnat i ng A E ~ Aa, dub, Ayb, andAuo i n Equat i ons ( 3) t hrough (lo), and combi ni ngXdc and xc i nto one arr ay.

Model s of vi rt ual l y al l the dc l i nks i n servi ce i nCanada and the Uni t ed Stat es have been const r uctedusi ng t he user defi ned model l i ng f aci l i t y [lo].

STATI C VAR COMPENSATOR

To date, sever al t ypes of SVCs have been used i n powersyst ems such as t hyri st or- cont r ol l ed r eact or ( TCR) ,t hyri st or- sw t ched capaci t or (TSC), and saturatedreactor ( SR) [6]. I n order t o provi de t he TCR orSR compensator s w t h a r eact i ve power product i onrange, capaci t i ve component s i n t he f orm of f i xedcapaci t or ( FC), mechani cal l y-sw t ched capaci t or, orTSC, are f r equent l y connected i n paral l el w t h t hemSi nce l i near system f ormul ati on does not al l ow f ordi scret e f unct i ons to be represent ed, al l f orms ofcapaci t i ve components ar e assumed f i xed, w t h t he f ast

act i on of TSC nor mal l y assumed i nstantaneous and t hus,i ncl uded i n t he TCR r ange. A TCR, on t he ot her hand,pr esent s a var i abl e suscept ance w t h f i r i ng angl econt r ol , whi ch may be provi ded w t h suppl ement arycont r ol s t o i mprove the dampi ng o f l ow f requencyosci l l at i ons [7]. Mor eover , an SR compensat or has anatural character i s t i c w th no f i ri ng angl e contr ol ,whi ch, as f ar as a smal l si gnal model i s concerned,may be i nt erpreted as a si mpl i f i ed f orm of TCR. As ar esul t , al l SVCs that are i n ser vi ce can be model l edas TCR+FC compensat or s.

Al t hough t he basi c power ci r cui t of a TCR i s st andard,t he met hod of i t s cont r ol and the nat ure of t he whol ecompensat or cont r ol var i abl es used var i es f r omi nst al l ati on to i nst al l at i on. The model present edhere i ncl udes fl exi bl e contr ol l er represent at i ons,w t h a general model l i ng f or m si ml ar to t hat of HVdcl i nk ( i e, Equat i ons 1) and 2 ) ) . Si m l ar l y, t hemodel i s f ormed usi ng t he component connect i on met hod,w t h t he normal cont i nuous appr oxi mat i on t o t he TCRval vi ng acti on, l i neari zed about an i ni ti al condi t i ondeterm ned f rom an i ni t i at i ng l oad f l ow TheSVC model of SSSP w t h user def i ned cont r ol s i s showni n Fi gur e 2 . The package has al so a model w t h asi mpl e cont rol l er t hat i s di scussed l ater.

Power Ci r cui t Model s

The power ci r cui t of an SVC i s model l ed as a vari abl esuscept ance, r esul t i ng i n the f ol l ow ng rel ati onshi p:

A svc = csvc A %VC+ ysvc A vsvc

wher e,

CSVC i s an arr ay descr i bi ng t he i nf l uence of t heSVC suscept ance on t he SVC cur r entchange vect or Ai svc,

YSVC i s t he eff ect i ve sel f bus adm t t ancematr i x, and,

AVSVC i s t he SVC sel f bus vol t age changevector.

LOCAL VOLTAGE P.F.INPUTS

REGVIATOR1 ND

MODULATION

k clUSERDEFNED COMRCI LWNL SVC MODELS

CCNTR~LS WUT CfTloN

FIGURE2 SVC (TCR + FC) USER DEFINED MODEL



1150

The change i n SVC susceptance i s t he same as t hechange i n TCR suscept ance AB^ -^ as the capaci t i vecomponent i s assumed f i xed. The TCR suscept ance i s anon- l i near f unct i on of t he TCR f i r i ng angl e a [ 6 1 :

2x - 2 a + sin 2x - 2 a )= B,

R

wher e, BL i s t he susceptance of t he r eact or used i nTCR .

Li near i zati on of Equati on ( 12) yi el ds:

A BbVC = A Bm = F a ) Aa

wher e,

F( a = - 28, 1 cos 2 a )/ E ( 1 4 )

The non- l i near SVC model of Fi gur e 2 t akes Equati on( 14) i nt o account , wher eas f or t he l i near SVC, t he TCRnon- l i neari ty i s cancel l ed by an i deal l i neari zi ngbl ock, maki ng F(a) a const ant equal t o - 2BL/ WEi t her model may be chosen, i n user- def i ned contr olopt i on. The l i near opt i on i s i nher ent i n t he si mpl eSVC model . The out put of t he user - def i ned contr ol i st he TCR fi r i ng angl e i n per uni t , wher e,

a - (1 5)a r a d - x / 2

PU XI2

Cont r ol Model s

Si m l ar t o HVdc, t wo t ypes of contr ol model s have beenpr ovi ded. The i ni t i al pl anni ng model i s based ont ypi cal cont r ol s, where t he cont r ol i nput can beei t her t he SVC sel f bus vol t age magni t ude or t hevol t age magni t ude of a r emote bus w t hi n t he acnetwork. I t s f ormul at i on i s si m l ar to that of t heuser - defi ned cont rol model t hat i s di scussed bel ow

User - Def i ned Cont r ol Model

Basi c l i near bui l di ng bl ocks, si ml ar to those ofHVdc, ar e avai l abl e f or model const r uct i on. Thei nt erf ace to t he rest of t he systemi s through vol t agemagni t ude, curr ent magni t ude, and r eact i ve power ofboth t he TCR and the whol e SVC, and thr ough si m l arquant i t i es associ ated w t h t wo r emote busesr epr esent i ng an ac l i ne (or t ransf ormer) , as wel l asr emot e bus phase angl e, and power and power f act orassoci at ed w t h the l i ne.

The f ormul at i on of t he cont rol s i s si ml ar toEquat i ons 6) t hrough 10) of t he HVdc model , exceptt hat t he fi r st and the l ast t erms of Equati on 10) donot exi st f or t he SVC model ( i e, CG and BG are zero).The gener al f orm of t he model i s obt ai ned byel i mnat i ng Aa, Aub, Ayb andduo among Equat i ons U), 1 3 ) , and modi f i ed 6)t hrough 10) .

EXAMPLES OF USE

Ei genval ue Anal ysi s

The capabi l i t i es and appl i cat i on of t he smal l si gnaldc l i nk and SVC model s ar e demonst r at ed by consi der i nga smal l ac/ dc syst em havi ng 10 buses, 6 ac l i nes, 5t r ansf ormers, t hr ee detai l ed gener ator model s, t wocl assi cal generator model s, one two-t erm nal dc l i nk,and one SVC at t he i nvert er bus. A si ngl e l i nedi agr amof t he syst emi s shown i n Fi gur e 3 .

Four cases have been st udi ed. The f i r st caserepresent s t he dc l i nk i n a form as cl ose to af unct i onal model as i s possi bl e. Thi s i s done byf orci ng const ant dc curr ent and vol t age by l arge dcl i ne i nductance and capaci t ance. The changes of t heconvert er. f i ri ng angl es are then l i neari zed w t hr espect t o t he changes of t hei r r especti ve ac bus

RGURE 3 SINGLE LINEDIAGRAMOF THE SYSTEM

vol t ages. The second case i s for a same syst em butw t h the detai l ed dc cont r ol s and act ual dc networkdynamcs r epr esent ed. The t hi r d case i s f or t hedynam c dc model augmented by dc cur r ent modul at i onder i ved fr om t he f r equency change at bus 5. Thef our t h case i s si m l ar t o the t hi rd but w t h t he SVCmodul ati on. The syst em dat a and cont r ol bl ockdi agr ams ar e presented i n Appendi x A .

The syst em i s smal l enough f or al l ei genval ues,

ei genvect ors and i nverse ei genvector s t o be cal cul atedusi ng t he Mul t i - Area Smal l Si gnal st abi l i t y pr ogram MASS) [3], an opt i on of SSSP. However , onl y t her otor angl e modes associ ated w t h the synchronousgener at ors ar e consi der ed her e. Tabl e 1 gi ves t hedampi ng rat i o, f r equency of osci l l ati on and t hemachi ne havi ng t he most par t i ci pat i on [ ] f or t her otor angl e modes of t he above f our st udy cases.

The f i r st t wo modes ( corr espondi ng t o DG1 and DG3) arer el at i vel y wel l damped and r emai n more or l ess t hesame i n al l cases. The thi r d and t he f our t h modes( corr espondi ng t o CG4 and CG5) are r ather l i ghtl ydamped w t h some di f f erences dependi ng on t he t ype oft he dc model used. I t can be seen t hat , t he

model has pr oduceduncti onal dc l i nkpessi m st i c r esul t f or t he t hi rd mode, andopt i m st i c r esul t f or t he f ourt h mode. The modul at i onof ei t her dc or SVC cont r ol s i ncreases t he dampi ngr ati os of t hese modes, w t h more i ncr ease f or t hef our t h mode t han f or t he t hi r d mode. Thi s i s due t ot he hi gher r el ati ve ampl i t ude of t he f our t h mode i nt he modul ati on si gnal t hat i s evi dent i n t he t i mesi mul ati on resul t s present ed next .

TABLE1 ROTORANGLE MODESFROMEIGENVALUEANALYSIS(MASS)

7 77



1151

4

Compar i son w t h Ti me Si mul at i on Resul t s

To veri f y the smal l si gnal stabi l i t y resul ts, ti mesi mul at i ons of t he same st udy cases have been car r i edout usi ng versi on 2. 0 of t he Ext ended Transi ent andM dter mSt abi l i t y Program ( ETMSP) devel oped by Ontar i oHydr o under EPRI RP1208-9[ 9]. ETMSP cont ai ns t he HVdcmodel devel oped by EPRI under RP1964-4. Thi s modelhas been extensi vel y t ested and veri f i ed by Mani t obaHVdc Resear ch Cent r e [2]. The change i n the measuredf r equency at bus 5 ( i e, t he si gnal used f o rmodul ati on) i s shown i n Fi gur e 4, f or t he f our st udycases. I n each case, t he modes have been exci t ed by

si mul ati ng a smal l ( t emporary) di st urbance i n the dcl i nk ( f rom 0. 05 t o 0. 1 second). Whi l e t he shapes ofthe pl ot s i n Fi gur e 4 are i n agr eement w t h the smal lsi gnal anal ysi s r esul t s, a cl oser compar i son can bemade by a spect r al anal ysi s of t hese cur ves: t her esul t s are pr esent ed i n Tabl e 2.

I I u u u u

svc ~ ~ , j ~ i ~ t i ~ ~.0125 I 2.5934 0.1 160.2560 1.4522 1.000

a i d svc Models

FIGURE 4 TIME SIMULATION RESULTS

-w -1'40.6-

-180 .6 -

-220.

- 3 o o . e

W

2 6 0 . 6

1

- 3 ' 40 .0 -

- 3 8 0 . 0

TABLE2 ROTORANGLEMODESFROMSPECTRALANALYSISO TIMESIMULATDN (ETMSP)OF FREQUENCY AT BUSNO. 5

Fr equency1 HZ.

10

RELATIVE

0.000Functional DC - _ - -

The r el ati ve ampl i t ude col umn of Tabl e 2 expl ai ns whyt he modul ati on cont r ol s have had vi r t ual l y no ef f ecton the f i r st t wo modes and l e ss ef f ect on t he t hi rdmode t han on t he f our t h mode. Vi r t ual l y zer o rel ati veampl i t udes of t he f i r st t wo modes al so poi nt t o thef act t hat t hese modes are hi ghl y damped, as t hespectr al anal yses have been perf ormed f or a t i me

w ndow st art i ng at 0. 4 seconds and endi ng at1. 5 seconds.

Fr equency Response Capabi l i t y

The SSSP package has al so a f r equency r esponsecapabi l i t y t hat i s very usef ul i n desi gni ngcont r ol l ers f or any devi ce. I n part i cul ar, thef r equency r esponse due to di st urbances at di f f erentpoi nts i n the contr ol s of dc l i nk and SVC model s maybe obtai ned. An exampl e i s pr ovi ded i n Fi gur e 5,wher e f or a l arge system cont ai ni ng a dc l i nk i nparal l el w t h an ac l i ne, t he gai n and Phase angl e oft he r at i o of t he power f l ow change t hrough t he ac l i net o a change i n the dc cur r ent or der are presented.The r esul t s i ndi cat e a mode at about 0. 35 Hz, whi chcoul d be cont r ol l ed by a modul at i on si gnal based ont he ac ti e l i ne power f l ow

:: - 1 0 .20.j-100 .



1152

Torsi onal I nt er acti ons

The detai l ed gener at or model of SSSP Package has amul t i - mass represent at i on capabi l i t y. I t pr ovi des fort he study of t orsi onal i nt eracti ons between machi neand HVdc l i nk cont r ol s. However, t he package i s notsui t abl e f or subsynchr onous r esonance and harmoni ci nt eracti on studi es. For such studi es, expl i ci tr epr esentat i on of t he network dynamcs i s essent i al[l l ]. HVdc l i nk and SVC model s whi ch i ncl ude t hedi scr ete nature of the f i r i ng system are al sor equi r ed; curr ent l y such model s ar e not avai l abl e for

smal l s i gnal s tabi l i ty anal ysi s .

CONCLUSI ONS

Because of t he cont r ol l abl e natur e of HVdct r ansm ssi on l i nks and SVCs, and thei r pot ent i al f oraf f ecti ng system stabi l i t y, there i s a need f or t hei rr epr esent ati on i n smal l si gnal stabi l i t y programs aswel l as t i me si mul at i on progr ams. Model s of bot hdc l i nks and SVCs f or use i n t i me s i mul at i on progr amshave been avai l abl e f or some t i me. I n t hi s paper, wehave addressed the for mul ati on of dc l i nks and SVCsand thei r contr ol l ers f or smal l si gnal stabi l i ty.Such model s, w t h bot h st andar d and user - def i nedcont r ol opt i ons, are avai l abl e i n t he SSSP packagedevel oped by Ontar i o Hydro under EPRI RP2447- 1.

I n order t o showt he capabi l i t y and appl i cat i on of t hesmal l si gnal dc l i nk and SVC model s, sever al caseshave been st udi ed usi ng t he MASS opt i on of t he SSSPpackage. Our st udi es show t hat a f unct i onal dc modelproduces t he f r equency and dampi ng of l ow f r equencymodes f ai r l y accurat el y. However , f or some modes thedampi ng i s l ower and for ot her s hi gher t han thatcal cul ated usi ng a f ul l dc model .

The f l exi bi l i t y of user- defi ned model has beendemonst r ated by i t s appl i cat i on t o modul ati on schemesf or both dc l i nks and SVCs. The f r equency r esponsecapabi l i t y of t he package, as r el ated to t hese model s,has al so been demonst r at ed.

Smal l si gnal stabi l i t y r esul t s of t he study cases havebeen ver i f i ed by t i me si mul ati on r esul t s of t he samecases. Cl ose agr eement bet ween t he t wo set s ofr esul t s has been demonst r ated by a spect r al anal ysi sof t he ti me si mul ati on resul t s.

ACKNOWLEDGEMENTS

The devel opment of t he SSSP package was f undedby EPRI ,under r esearch pr oj ect RP2447-1.

REFERENCES

Compendi um of HVDC schemes Throughout t heWorl d , I nt ernat i onal conference on Large Hi ghVol t age El ect r i c Syst ems, CI GRE WG 04 of SC 14,1987.

Methodol ogy f or t he I ntegr ati on of HVDC Li nks i nLar ge AC Syst ems - Phase 2: Advanced Concept s ,EPRI EL- 4365 CCM RP 1964- 2 User ' s Manual ,Apri l 1987.

P. Kundur, G J . Rogers , D Y. Wong, L. Wang andMG Lauby, A Compr ehensi ve Comput er Programf or Smal l Si gnal St abi l i t y Anal ysi s of PowerSyst ems , 9OWM 007- 5, I EEE PES Wnter Meet i ng,At l ant a, Februar y 1990.

R. A. DeCar l o, and R Saeks, I nter connect edDynam cal Syst ems, Marcel Dekker , I nc. , 1981.

E. W Ki mbark, D r ect Curr ent Transmssi on,W l l ey, New York, 1971.

T. J . E. M l l er, Reacti ve Power Contr ol i n El ectr i cSyst ems, J ohn Wl l ey Sons, 1982.

E. V. Larsen, J . H Chow SVC Cont r ol Desi gnConcept s f or Syst em Dynam c Per f ormance , I EEESpeci al Symposi um on Appl i cat i on of SVS f or,Syst em Dynam c Perf ormance, publ i cati on 87 THO187- 5- PWR, 1987.

I . J . Per ez-Arr i aga, GC Verghese, F. C. Scheppe,Sel ect i ve Modal Anal ysi s w t h Appl i cat i ons t o

El ect r i c Power Systems , I EEE Tr ansact i on onPower Apparatus and Syst ems, Vol . PAS- 101, pp.3117- 3134, 1982.

Extended Transi ent - M dterm St abi l i t y ProgramPackages Vers i on 2. 0 , EPRI EL-6648, RP 1208- 9User Manual , December 1989.

[ l o] Wor ki ng Gr oup on HVDC Model l i ng f or Stabi l i t ySi mul at i ons , NPCC- SS34 WG, Par t I 11 Report(Draft ) , 1990.

[ l l ] G Gr oss, C. F. I mparato and MP. Look, A Toolf or t he Comprehensi ve Anal ysi s of Power Syst emDynamc St abi l i t y , I EEE Tr ans, PAS- 101, 1981,pp 226, 234.

APPEND X A

The data of t he study system ( Fi gure 3) i s as f o~ ows:

Load Fl ow Bases

100. 0 MVA20. 0 kV f or buses 1 t hrough 5500.0 kV f or buses 6 t hrough 10

AC Li nes

Li ne R ( pu) XQ ( pu) Bc ( pu)1 0.0 0. 001 0. 02 0. 003 0. 03 8. 03 0. 006 0. 06 4 . 0

4 0. 003 0.02 2.00. 06 8.06 0. 006

Tr ansf ormers

XQ = 0.006 pu f or T1 t hrough T3XQ = 0. 004 pu fo r T4 and T5

CaDac o r s

C1 = 6.0 puc2 = 7. 4 pu

20. 0+ 7. 0 pu cont . MVA f or l oads 1 & 210. 0+ 3. 0 pu cont . MVA f or l oad 3

0. 0- j 4. 0 pu cont. i mpedance f or l oad 3

Detai l ed Generat ors

Ra=O O xd=1. 94 PU Xq=1. 91 XQ=o. 16 pU,

x qo 19 U T' do=7. 3 S T' qo=0. 7 S I

T do=0. 033 S T qo=0. 06 S, H=3. 8 MWS/ MVA, D=1. 0pu of t or que/ pu of speed devi at i on, Based050 MVA.

I EEE Type' DC1 ( DC Commutat or) Exci t er [91:

K~=19. 0, T*=O 15, KE=~. , T~=0. 65,AEX=0. 000003387, B~x=2. 1972, K~=0. 05, T~=2. 0,

x' dz0. 29 PU, ~' ~=0 . 4 6 US p ; ' *d=o. 19 pus

VRm=2. 5# VRM N= 2. 5, T~=0. 06.

r 7 -

Análisis de Estabilidad de Pequeña Señal Para SVC y HVDC

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